So far, it has been assumed that each Kohn-Sham eigenstate has been fully occupied. That is, each state has associated with it an electronic charge for each point in the Brillouin zone sampling which contributes to the total energy of the system. This is not true for metals. In metals there are partially filled bands because the Fermi surface cuts through some bands.
In order to find the Fermi surface it is necessary to sample part of the unoccupied band structure because of the partial occupancy of some bands. This is done by weighting each state near the (estimated) Fermi surface by the occupied portion of the reciprocal space volume it represents. An effective technique of doing this is Gaussian smearing which is a method that has been also used to treat electrons at non-zero temperatures. In this scheme, the energy of each calculated band is broadened by a gaussian. That is to say, the weight (or occupation number) is chosen by the portion of the gaussian distribution which lies under the Fermi level. The Gaussian is given by
where is the energy of band i and k-point , is the Fermi energy and is the width of the Gaussian which is chosen appropriate to the spacing between the energy bands. It is initially chosen large and gradually reduced as the convergence to the Fermi surface improves.
As stated above, the occupation number is chosen by the area under the Gaussian that lies below the Fermi surface:
where erf(x) is the standard error function and normalisation gives
where is the total number of electrons in the unit cell.
In this method using a generalisation of density functional theory to non-zero temperatures, it is more useful to work in a free energy formalism which depends on both the wavefunctions and the single particle occupation numbers
which introduces an entropy correction
to the total energy of the system. Clearly as , the entropy goes to zero and the usual zero temperature scheme is recovered. This addition of the entropy term to the standard energy functional makes it possible to treat a system with fractional occupancies within a global minimisation procedure with respect to the wavefunction coefficients and the occupation numbers.
It is important that a large number of k-point is used to sample the Brillouin zone so that the Fermi surface is accurately mapped out. A failure to do so will result in a large error in the Fermi energy leading to incorrect part of the band structure to be occupied or empty.
This method is used in Chapter 3, where the the metallic phase, -Sn, of silicon is considered.